Subgroups — C₅₃×#19277 #27226789

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3079	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3079	C₂	1	1	yes	maximal, characteristic
4	C₈×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1684	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1683	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#487	—	2	1	no	—
14	C₅₃×#487	—	2	1	no	—
15	C₅₃×#487	C₂²	1	4	yes	—
16	C₅₃×#477	—	2	1	no	—
17	C₅₃×#477	C₂²	1	2	yes	—
18	C₅₃×#476	—	2	1	no	—
19	C₅₃×#476	C₂²	1	2	yes	—
20	C₅₃×#475	—	2	1	no	—
21	C₅₃×#475	C₂²	1	2	yes	—
22	C₅₃×#474	—	2	1	no	—
23	C₅₃×#474	C₂²	1	2	yes	—
24	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
25	C₈×D₄×C₅₃	—	2	2	no	—
26	C₈×D₄×C₅₃	C₂²	1	2	yes	—
27	C₅₃×#431	—	2	1	no	—
28	C₅₃×#431	C₂²	1	1	yes	characteristic
29	C₅₃×#428	—	2	1	no	—
30	C₅₃×#428	—	2	1	no	—
31	C₅₃×#423	C₂²	1	1	yes	characteristic
32	C₅₃×#423	C₂²	1	1	yes	characteristic
33	C₅₃×#417	—	2	1	no	—
34	C₅₃×#417	—	2	1	no	—
35	C₂×#103×C₅₃	—	2	2	no	—
36	C₂×C₄×C₈×C₅₃	—	2	1	no	—
37	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#335	C₂²	1	4	yes	—
39	C₅₃×#333	C₂²	1	4	yes	—
40	C₅₃×#332	C₂²	1	4	yes	—
41	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
42	C₂²×Q₈×C₅₃	—	2	1	no	—
43	C₂²×Q₈×C₅₃	—	2	1	no	—
44	C₂×M₁₆×C₅₃	—	2	1	no	—
45	C₂×M₁₆×C₅₃	—	2	1	no	—
46	C₂²×C₈×C₅₃	—	2	2	no	—
47	C₂²×C₈×C₅₃	D₄	1	2	yes	—
48	C₄⋊Q₈×C₅₃	—	2	1	no	—
49	C₄⋊Q₈×C₅₃	—	2	1	no	—
50	C₄⋊Q₈×C₅₃	—	2	1	no	—
51	C₄⋊Q₈×C₅₃	—	2	1	no	—
52	C₄⋊Q₈×C₅₃	—	2	1	no	—
53	C₄⋊Q₈×C₅₃	—	2	1	no	—
54	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
55	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
56	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
57	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
58	C₄×D₄×C₅₃	—	2	1	no	—
59	C₄×D₄×C₅₃	—	2	1	no	—
60	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
61	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
62	#117×C₅₃	—	2	1	no	—
63	#117×C₅₃	D₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
65	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
67	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
68	C₂×C₄²×C₅₃	—	2	1	no	—
69	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
70	#107×C₅₃	—	2	2	no	—
71	#107×C₅₃	—	2	2	no	—
72	#106×C₅₃	—	2	2	no	—
73	#106×C₅₃	—	2	2	no	—
74	C₄⋊C₈×C₅₃	—	2	2	no	—
75	C₄⋊C₈×C₅₃	—	2	1	no	—
76	C₄⋊C₈×C₅₃	—	2	1	no	—
77	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
78	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
79	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
80	#103×C₅₃	—	2	4	no	—
81	#103×C₅₃	—	2	4	no	—
82	#103×C₅₃	—	2	2	no	—
83	#103×C₅₃	—	2	2	no	—
84	#102×C₅₃	—	2	2	no	—
85	#102×C₅₃	—	2	2	no	—
86	C₂²⋊C₈×C₅₃	—	2	2	no	—
87	C₂²⋊C₈×C₅₃	—	2	1	no	—
88	C₄×C₈×C₅₃	—	2	2	no	—
89	C₄×C₈×C₅₃	—	2	1	no	—
90	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
91	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
92	C₄×C₈×C₅₃	C₂³	1	2	yes	—
93	#41×C₅₃	—	2	2	no	—
94	#41×C₅₃	Q₁₆	1	4	yes	—
95	C₂×Q₈×C₅₃	—	2	2	no	—
96	C₂×Q₈×C₅₃	—	2	2	no	—
97	C₂×Q₈×C₅₃	—	4	4	no	—
98	C₂×Q₈×C₅₃	—	4	1	no	—
99	C₂×Q₈×C₅₃	—	4	1	no	—
100	C₂×Q₈×C₅₃	—	8	2	no	—
101	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄×C₅₃	—	2	1	no	—
103	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	—	2	1	no	—
105	C₂²×C₄×C₅₃	—	2	1	no	—
106	C₂²×C₄×C₅₃	—	2	1	no	—
107	C₂²×C₄×C₅₃	—	4	1	no	—
108	C₂²×C₄×C₅₃	—	4	1	no	—
109	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
112	M₁₆×C₅₃	—	4	2	no	—
113	M₁₆×C₅₃	—	4	2	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	2	2	no	—
117	C₂×C₈×C₅₃	—	4	2	no	—
118	C₂×C₈×C₅₃	—	4	4	no	—
119	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
120	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
121	C₄⋊C₄×C₅₃	—	2	1	no	—
122	C₄⋊C₄×C₅₃	—	2	1	no	—
123	C₄⋊C₄×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	2	2	no	—
125	C₄⋊C₄×C₅₃	—	2	2	no	—
126	C₄⋊C₄×C₅₃	—	2	2	no	—
127	C₄⋊C₄×C₅₃	—	4	1	no	—
128	C₄⋊C₄×C₅₃	—	4	1	no	—
129	C₄⋊C₄×C₅₃	—	4	1	no	—
130	C₄⋊C₄×C₅₃	—	4	1	no	—
131	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
133	C₂²⋊C₄×C₅₃	—	2	1	no	—
134	C₂²⋊C₄×C₅₃	—	2	1	no	—
135	C₂²⋊C₄×C₅₃	—	2	1	no	—
136	C₄²×C₅₃	—	2	1	no	—
137	C₄²×C₅₃	—	2	1	no	—
138	C₄²×C₅₃	—	2	1	no	—
139	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
141	C₄²×C₅₃	#41	1	1	yes	characteristic
142	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
143	C₂³×C₅₃	—	2	1	no	—
144	C₂³×C₅₃	#121	1	1	yes	characteristic
145	Q₈×C₅₃	—	2	1	no	—
146	Q₈×C₅₃	—	4	4	no	—
147	Q₈×C₅₃	—	8	2	no	—
148	Q₈×C₅₃	—	8	2	no	—
149	Q₈×C₅₃	—	8	2	no	—
150	Q₈×C₅₃	C₂×Q₁₆	1	2	yes	—
151	D₄×C₅₃	—	2	4	no	—
152	D₄×C₅₃	—	2	1	no	—
153	D₄×C₅₃	C₂×Q₁₆	1	2	yes	—
154	C₂×C₄×C₅₃	—	2	1	no	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	2	4	no	—
157	C₂×C₄×C₅₃	—	2	2	no	—
158	C₂×C₄×C₅₃	—	2	1	no	—
159	C₂×C₄×C₅₃	—	2	1	no	—
160	C₂×C₄×C₅₃	—	4	2	no	—
161	C₂×C₄×C₅₃	—	4	2	no	—
162	C₂×C₄×C₅₃	—	4	2	no	—
163	C₂×C₄×C₅₃	—	4	2	no	—
164	C₂×C₄×C₅₃	—	8	2	no	—
165	C₂×C₄×C₅₃	—	8	2	no	—
166	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
167	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
168	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
170	C₂×C₄×C₅₃	#135	1	2	yes	—
171	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
174	C₈×C₅₃	—	2	2	no	—
175	C₈×C₅₃	—	2	2	no	—
176	C₈×C₅₃	—	4	2	no	—
177	C₈×C₅₃	—	4	2	no	—
178	C₈×C₅₃	—	4	2	no	—
179	#19277	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₂²×C₅₃	—	2	1	no	—
181	C₂²×C₅₃	—	2	1	no	—
182	C₂²×C₅₃	—	2	2	no	—
183	C₂²×C₅₃	—	4	1	no	—
184	C₂²×C₅₃	—	4	1	no	—
185	C₂²×C₅₃	#467	1	2	yes	—
186	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
187	C₄×C₅₃	—	2	1	no	—
188	C₄×C₅₃	—	2	2	no	—
189	C₄×C₅₃	—	2	1	no	—
190	C₄×C₅₃	—	4	1	no	—
191	C₄×C₅₃	—	4	1	no	—
192	C₄×C₅₃	—	8	2	no	—
193	C₄×C₅₃	—	8	2	no	—
194	C₄×C₅₃	#467	1	2	yes	—
195	C₄×C₅₃	#471	1	2	yes	—
196	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
197	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
198	C₄×C₅₃	#575	1	2	yes	—
199	C₂×#487	C₂×C₅₃	1	1	yes	characteristic, complemented
200	#3079	C₂×C₅₃	1	1	yes	characteristic
201	#3079	C₂×C₅₃	1	1	yes	characteristic
202	C₈×#41	C₂×C₅₃	1	1	yes	characteristic
203	#1684	C₂×C₅₃	1	4	yes	complemented
204	#1683	C₂×C₅₃	1	4	yes	complemented
205	#1567	C₂×C₅₃	1	1	yes	characteristic, complemented
206	#1549	C₂×C₅₃	1	1	yes	characteristic, complemented
207	#1548	C₂×C₅₃	1	1	yes	characteristic, complemented
208	C₂×C₅₃	—	2	2	no	—
209	C₂×C₅₃	—	4	2	no	—
210	C₂×C₅₃	#3040	1	1	yes	characteristic
211	C₂×C₅₃	C₂×#467	1	1	yes	characteristic
212	C₂×C₅₃	C₂×#471	1	1	yes	characteristic
213	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
214	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
215	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
216	#487	—	2	1	no	—
217	#487	—	2	1	no	—
218	#487	C₂²×C₅₃	1	4	yes	—
219	#477	—	2	1	no	—
220	#477	C₂²×C₅₃	1	2	yes	—
221	#476	—	2	1	no	—
222	#476	C₂²×C₅₃	1	2	yes	—
223	#475	—	2	1	no	—
224	#475	C₂²×C₅₃	1	2	yes	—
225	#474	—	2	1	no	—
226	#474	C₂²×C₅₃	1	2	yes	—
227	C₈×Q₈	C₂²×C₅₃	1	2	yes	—
228	C₈×D₄	—	2	2	no	—
229	C₈×D₄	C₂²×C₅₃	1	2	yes	—
230	#431	—	2	1	no	—
231	#431	C₂²×C₅₃	1	1	yes	characteristic
232	#428	—	2	1	no	—
233	#428	—	2	1	no	—
234	#423	C₂²×C₅₃	1	1	yes	characteristic
235	#423	C₂²×C₅₃	1	1	yes	characteristic
236	#417	—	2	1	no	—
237	#417	—	2	1	no	—
238	C₂×#103	—	2	2	no	—
239	C₂×C₄×C₈	—	2	1	no	—
240	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
241	#335	C₂²×C₅₃	1	4	yes	—
242	#333	C₂²×C₅₃	1	4	yes	—
243	#332	C₂²×C₅₃	1	4	yes	—
244	C₅₃	#19277	1	1	yes	characteristic, direct product factor, Sylow
245	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
246	C₂²×Q₈	—	2	1	no	—
247	C₂²×Q₈	—	2	1	no	—
248	C₂×M₁₆	—	2	1	no	—
249	C₂×M₁₆	—	2	1	no	—
250	C₂²×C₈	—	2	2	no	—
251	C₂²×C₈	D₄×C₅₃	1	2	yes	—
252	C₄⋊Q₈	—	2	1	no	—
253	C₄⋊Q₈	—	2	1	no	—
254	C₄⋊Q₈	—	2	1	no	—
255	C₄⋊Q₈	—	2	1	no	—
256	C₄⋊Q₈	—	2	1	no	—
257	C₄⋊Q₈	—	2	1	no	—
258	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
259	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
260	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
261	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
262	C₄×D₄	—	2	1	no	—
263	C₄×D₄	—	2	1	no	—
264	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
265	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
266	#117	—	2	1	no	—
267	#117	D₄×C₅₃	1	1	yes	characteristic
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	—	2	1	no	—
272	C₂×C₄²	—	2	1	no	—
273	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
274	#107	—	2	2	no	—
275	#107	—	2	2	no	—
276	#106	—	2	2	no	—
277	#106	—	2	2	no	—
278	C₄⋊C₈	—	2	2	no	—
279	C₄⋊C₈	—	2	1	no	—
280	C₄⋊C₈	—	2	1	no	—
281	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
282	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
283	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
284	#103	—	2	2	no	—
285	#103	—	2	2	no	—
286	#103	—	2	4	no	—
287	#103	—	2	4	no	—
288	#102	—	2	2	no	—
289	#102	—	2	2	no	—
290	C₂²⋊C₈	—	2	1	no	—
291	C₂²⋊C₈	—	2	2	no	—
292	C₄×C₈	—	2	2	no	—
293	C₄×C₈	—	2	1	no	—
294	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
295	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
296	C₄×C₈	C₂³×C₅₃	1	2	yes	—
297	#41	—	2	2	no	—
298	#41	Q₁₆×C₅₃	1	4	yes	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	4	4	no	—
302	C₂×Q₈	—	4	1	no	—
303	C₂×Q₈	—	4	1	no	—
304	C₂×Q₈	—	8	2	no	—
305	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
306	C₂×D₄	—	2	1	no	—
307	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	4	1	no	—
312	C₂²×C₄	—	4	1	no	—
313	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
314	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
315	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
316	M₁₆	—	4	2	no	—
317	M₁₆	—	4	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	4	2	no	—
322	C₂×C₈	—	4	4	no	—
323	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
324	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	1	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	4	1	no	—
332	C₄⋊C₄	—	4	1	no	—
333	C₄⋊C₄	—	4	1	no	—
334	C₄⋊C₄	—	4	1	no	—
335	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
336	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
337	C₂²⋊C₄	—	2	1	no	—
338	C₂²⋊C₄	—	2	1	no	—
339	C₂²⋊C₄	—	2	1	no	—
340	C₄²	—	2	1	no	—
341	C₄²	—	2	1	no	—
342	C₄²	—	2	1	no	—
343	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
344	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
345	C₄²	#41×C₅₃	1	1	yes	characteristic
346	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
347	C₂³	—	2	1	no	—
348	C₂³	#121×C₅₃	1	1	yes	characteristic
349	Q₈	—	2	1	no	—
350	Q₈	—	4	4	no	—
351	Q₈	—	8	2	no	—
352	Q₈	—	8	2	no	—
353	Q₈	—	8	2	no	—
354	Q₈	C₂×Q₁₆×C₅₃	1	2	yes	—
355	D₄	—	2	4	no	—
356	D₄	—	2	1	no	—
357	D₄	C₂×Q₁₆×C₅₃	1	2	yes	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	8	2	no	—
369	C₂×C₄	—	8	2	no	—
370	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
371	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
372	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
373	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
374	C₂×C₄	#135×C₅₃	1	2	yes	—
375	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
376	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
377	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
378	C₈	—	2	2	no	—
379	C₈	—	2	2	no	—
380	C₈	—	4	2	no	—
381	C₈	—	4	2	no	—
382	C₈	—	4	2	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	2	no	—
386	C₂²	—	4	1	no	—
387	C₂²	—	4	1	no	—
388	C₂²	C₅₃×#467	1	2	yes	—
389	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
390	C₄	—	2	1	no	—
391	C₄	—	2	2	no	—
392	C₄	—	2	1	no	—
393	C₄	—	4	1	no	—
394	C₄	—	4	1	no	—
395	C₄	—	8	2	no	—
396	C₄	—	8	2	no	—
397	C₄	C₅₃×#467	1	2	yes	—
398	C₄	C₅₃×#471	1	2	yes	—
399	C₄	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
400	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
401	C₄	C₅₃×#575	1	2	yes	—
402	C₂	—	2	2	no	—
403	C₂	—	4	2	no	—
404	C₂	C₅₃×#3040	1	1	yes	characteristic
405	C₂	C₂×C₅₃×#467	1	1	yes	characteristic
406	C₂	C₂×C₅₃×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂×#41×C₅₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₅₃×#487	C₂	semidirect
5. C₅₃×#1684	C₂	semidirect
6. C₅₃×#1683	C₂	semidirect
7. C₅₃×#1567	C₂	semidirect
8. C₅₃×#1549	C₂	semidirect
9. C₅₃×#1548	C₂	semidirect
179. #19277	C₅₃	direct (N × H)
199. C₂×#487	C₂×C₅₃	semidirect
203. #1684	C₂×C₅₃	semidirect
204. #1683	C₂×C₅₃	semidirect
205. #1567	C₂×C₅₃	semidirect
206. #1549	C₂×C₅₃	semidirect
207. #1548	C₂×C₅₃	semidirect
244. C₅₃	#19277	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅₃×#487	2
C₅₃×#3079	2
C₈×#41×C₅₃	2
C₅₃×#1684	2
C₅₃×#1683	2
C₅₃×#1567	2
C₅₃×#1549	2
C₅₃×#1548	2
#19277	53

Subgroups — C53×#19277 #27226789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19277 #27226789